Analysis Proof Help :S

Hi, im taking my first analysis course and we are studying Limits right now. My prof said they are the most important thing to remember out of this whole semester. Anyways i have two problems im trying to solve that i could do with some help.

1) Show that if [tex]f(x) \leq 0[/tex] and [tex]\lim_{x->a} f(x) = l[/tex], then [tex]l \leq 0[/tex].

2) If [tex]f(x) \leq g(x)[/tex] for all x, then [tex]\lim_{x->a} f(x) \leq \lim_{x->a} g(x)[/tex].
If those limits exist.

For number one i can see this is obvious but i don't know where to start to try and prove it.

I know the definitions for limits, do i use them somehow?

For number 1) i think that i can use a proof by contradiction somehow.

however, the next steps are wrong. For instance, the distance from [itex]f(x)[/itex] to zero could, in fact, be zero. (e.g. take [itex]f(x) = 0[/itex], or [itex]f(x) = -|x|[/itex])

I'm also not sure what you're trying to do when you say "choose [itex]\delta[/itex] such that [itex]\epsilon = \alpha[/itex]"... when deriving the contradiction you do get to choose [itex]\epsilon[/itex], but the contradiction must hold for all [itex]\delta[/itex].

You need another hint on this one?

As for number two, the proof you gave almost works, now that you've proven #1... see if you can rewrite it to take advantage of knowing #1.